= Directional error of a first-order Taylor approximation
{title2=$f(a+sv)-f(a)-sDf(a)v=\tfrac12s^2v^TH_f(a)v+O(s^3)$}
For a fixed unit direction $v$, the <multivariable Taylor expansion> gives the displayed leading error. Equal <directional derivatives> in two directions do not imply equal approximation errors: the linear changes depend on the <gradient>, while the quadratic errors depend on the <Hessian matrix>. For example, $f(x,y)=x+y+2x^2+y^2$ at the origin has equal derivatives in the two coordinate directions but errors $2s^2$ and $s^2$.
Back to article page
